6174 is known as Kaprekar's constant after the Indian mathematician D. R. Kaprekar. This number is notable for the following property:Here's a screenshot of the GeoGebra applet:
The above process, known as Kaprekar's routine, will always reach its fixed point, 6174, in at most 7 iterations. Once 6174 is reached, the process will continue yielding 7641 – 1467 = 6174. For example, choose 3524:
- Take any four-digit number, using at least two different digits. (Leading zeros are allowed.)
- Arrange the digits in ascending and then in descending order to get two four-digit numbers, adding leading zeros if necessary.
- Subtract the smaller number from the bigger number.
- Go back to step 2.
The only four-digit numbers for which Kaprekar's routine does not reach 6174 are repdigits such as 1111, which give the result 0 after a single iteration. All other four-digit numbers eventually reach 6174 if leading zeros are used to keep the number of digits at 4:
- 5432 – 2345 = 3087
- 8730 – 0378 = 8352
- 8532 – 2358 = 6174
Saturday, November 12, 2011
Kaprekar's Constant (6174) in GeoGebra
While doing a little browsing, I came across Kaprekar's Constant. I remember reading about it a few years ago but I never did anything with it. I finally decided to play with it a little using GeoGebra. Here's a description from Wikipedia: